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Modern 4DVar

Spatio-Temporal Data Representation

What components can we use to estimate the state?

CNRS
MEOM

Core Data Structures

Points

Lines

Polygons

Rasters

Spatiotemporal Dependencies

Indepedence

Spatial Independence

Here, we assume that there is no dependence on the spatial coordinates.

Ω={xn∈Ω⊆RDs}n=1Ns\boldsymbol{\Omega} = \left\{ \mathbf{x}_n\in\boldsymbol{\Omega}\subseteq\mathbb{R}^{D_s}\right\}_{n=1}^{N_s}
(1)

We can stack this vector which includes all of the spatial coordinates within the domain.

X=[x1,x2,…,xNs]∈RNs×Ds\mathbf{X} = [\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_{N_s}] \in \mathbb{R}^{N_s\times D_s}
(2)

Temporal Independence

T={tn∈T⊆R+}n=1Nt\mathcal{T} = \left\{ t_n\in\mathcal{T}\subseteq\mathbb{R}^+\right\}_{n=1}^{N_t}
(3)

We can stack this vector which includes all of the temporal coordinates within the domain.

T=[t1,t2,…,tNs]∈RNt\mathbf{T} = [t_1, t_2, \ldots, t_{N_s}] \in \mathbb{R}^{N_t}
(4)

Dependence

Spatial Dependence

Here, we assume that there is full dependence on

Ω={xd∈Ω⊆RDs}d=1DΩ\boldsymbol{\Omega} = \left\{ \mathbf{x}_d\in\boldsymbol{\Omega}\subseteq\mathbb{R}^{D_s}\right\}_{d=1}^{D_\Omega}
(5)

We can stack this vector which includes all of the spatial coordinates within the domain.

X=[x1,x2,…,xDΩ]∈RDΩ×Ds\mathbf{X} = [\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_{D_\Omega}] \in \mathbb{R}^{D_\Omega\times D_s}
(6)

Temporal Dependence

Here, we assume that there is no dependence

T={td∈[0,T]⊆R+}d=1Dt\mathcal{T} = \left\{t_d \in [0,T]\subseteq\mathbb{R}^+ \right\}_{d=1}^{D_t}
(7)

We can stack this vector which includes all of the temporal coordinates within the domain.

T=[t1,t2,…,tDt]∈RDt\mathbf{T} = [t_1, t_2, \ldots, t_{D_t}] \in \mathbb{R}^{D_t}
(8)

Partial Dependence

In many cases, there is assumed to be partial dependence. This is reasonable for many geospatial variables because we can assume that the nearby neighbours are

Partial Spatial Dependence

Here, we assume that there is partial dependence on the spatial variables. So first, we partition the space into pp subdomains.

Ω={Ωp⊆Ω}p=1Np\boldsymbol{\Omega} = \left\{ \boldsymbol{\Omega}_p\subseteq \boldsymbol{\Omega}\right\}_{p=1}^{N_p}
(9)

Notice that we have NpN_p samples for this set and not DpD_p. This is because we assume that there is no dependence between the partitions. However, we can do some hierarchical dependencies

Now, we do the same as above for the spatial dependence. However, we can apply the spatial dependence set on each of the partitions.

Ωp={xd∈Ω⊆RDs}d=1DΩ\boldsymbol{\Omega}_p = \left\{ \mathbf{x}_d\in\boldsymbol{\Omega}\subseteq\mathbb{R}^{D_s}\right\}_{d=1}^{D_\Omega}
(10)

We can stack this vector which includes all of the spatial coordinates within the domain.

Xp=[x1,x2,…,xDΩp]∈RDΩp×Ds\mathbf{X}_p = [\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_{D_{\Omega_p}}] \in \mathbb{R}^{D_{\Omega_p}\times D_s}
(11)

We can stack all of the variables together to include all of the spatial coordinates for all of the domains.

X=[X1,X2,…,XNp]∈RNp×DΩp×Ds\mathbf{X} = [\mathbf{X}_{1}, \mathbf{X}_{2}, \ldots, \mathbf{X}_{N_p}] \in \mathbb{R}^{N_p\times D_{\Omega_p}\times D_s}
(12)

Partial Temporal Dependence

Here, we assume that there is partial dependence on the temporal variables. So first, we partition the space into pp subdomains.

T={Tp⊆T}p=1Np\boldsymbol{\mathcal{T}} = \left\{ \boldsymbol{\mathcal{T}}_p\subseteq \boldsymbol{\mathcal{T}}\right\}_{p=1}^{N_p}
(13)

Notice that we have NpN_p samples for this set and not DpD_p. This is because we assume that there is no dependence between the partitions. However, we can do some hierarchical dependencies

Now, we do the same as above for the temporal dependence. However, we can apply the spatial dependence set on each of the partitions.

Tp={td∈T⊆R+}d=1DTp\boldsymbol{\mathcal{T}}_p = \left\{ t_d\in\boldsymbol{\mathcal{T}}\subseteq\mathbb{R}^+\right\}_{d=1}^{D_{\mathcal{T}_p}}
(14)

We can stack this vector which includes all of the spatial coordinates within the domain.

Tp=[t1,t2,…,tDΩp]∈RDTp×Ds\mathbf{T}_p = [t_1, t_2, \ldots, t_{D_{\Omega_p}}] \in \mathbb{R}^{D_{\mathcal{T}_p}\times D_s}
(15)

We can stack all of the variables together to include all of the spatial coordinates for all of the domains.

T=[T1,T2,…,TNp]∈RNp×DTp\mathbf{T} = [\mathbf{T}_{1}, \mathbf{T}_{2}, \ldots, \mathbf{T}_{N_p}] \in \mathbb{R}^{N_p\times D_{\mathcal{T}_p}}
(16)
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